Deriving Field Equations of Einstein-Cartan Theory from Hilbert Action

[tetraedro]

I'm interested in the derivation of the field equations of Einstein-Cartan theory of gravity starting from the Hilbert action, but I can't find a book or an article which explicitly derives the equation satisfied by the torsion field using the variation of the action with respect to the spin connection. Can you suggest me please something which could be of any help?
Thank you

 

[zorgkang]

I was going to post almost the exact same question. Maybe a bit easier, where can I find a good introduction to the tetrad formalism? Hopefully including the information asked for above. Cheers

 

[zorgkang]

Hehe, interesting response. I don't really see why though. As I understand it, Einstein-Cartan is just the extension of GR allowing for non-zero torsion and is derived from the same Hilbert-Einstein action, but taking the connection instead of the metric as dynamical variables. (might be slightly wrong here, this is what I'm trying to better understand) Anyway, it's a perfectly sound theory and the natural extension to GR including matter with non zero spin.

That being said, I am happy to answer your questions. I am a physics grad student, and right now I am trying to better understand the implications of varying the metric or the connection coefficients or both or the tetrad, or spin connection, etc. I remember that I started reading about the tetrad formalism last year, but I can't remember where and I'm just asking for help finding a good source since the couple standard GR references I've looked at don't seem to use tetrad formalism.

Do I pass the crazy test? just kidding. ;)

 

[pmb_phy]

Hehe, interesting response. I don't really see why though. As I understand it, Einstein-Cartan is just the extension of GR allowing for non-zero torsion and is derived from the same Hilbert-Einstein action, but taking the connection instead of the metric as dynamical variables. (might be slightly wrong here, this is what I'm trying to better understand) Anyway, it's a perfectly sound theory and the natural extension to GR including matter with non zero spin.

Have you tried searching the internet for this? I could give you a hand searching if you'd like. Just say the word.

That being said, I am happy to answer your questions. I am a physics grad student, and right now I am trying to better understand the implications of varying the metric or the connection coefficients or both or the tetrad, or spin connection, etc. I remember that I started reading about the tetrad formalism last year, but I can't remember where and I'm just asking for help finding a good source since the couple standard GR references I've looked at don't seem to use tetrad formalism.

Do I pass the crazy test? just kidding. ;)

Please ignore all references to questions on motivations for your question. Especially if their are personal reasons for doing so. I find it rude that someone should ask this. That kind of question belongs in PM since that is one of the things that PM was created for. Personally I find nothing strange about you're question.

Best wishes

Pete

 

[zorgkang]

Thanks, well yes I've been reading here and there all afternoon. Read some stuff on wikipedia, a few articles, forum posts, looked in standard GR books, etc. But what I really need is a book or review article or something like that. I suppose I should also say I'm looking for 2 things:

1. the tetrad formalism
I'm sure there are some good introductions on that. I guess I'm unlucky or lack some searching skills.

2. Einstein-Cartan theory, and its derivation from an action
I think this is usually presented in the tetrad formalism, but obviously is a more specific thing and possibly harder to find.

Anyway, I am still reading and searching. I just tought since I ran across this question without an answer that I'd bring it back up. So if anyone *knows* of references that might help, please let me know.

(edit: written before the last post appeared. thanks.)

 

[zorgkang]

Well, see I am totally unaware and uninterested in all that. Maybe you should relax and bring up such things only when people are acutally talking about it. From the abstract you linked, that "theory" used Einstein-Cartan geometry and that is the only link to this thread. Our questions were *clearly* not related. This is not the place for cranks, but I think it is not the place for crank-hunting either.

I'll take a look at your first link. thanks

 

[Chris Hillman]

Two citations which might help

A review of Einstein-Cartan theory:
http://www.arxiv.org/abs/gr-qc/0606062
For some historical background, see
http://relativity.livingreviews.org/About/authors.html#goennerhubert

For a nice review of tetrads, try: Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt, Exact Solutions of Einstein's Field Equations, 2nd ed.,
Cambridge monographs on mathematical physics; no. 6., Cambridge University Press, 2003.

Enjoy!

 

[Federation 2005]

An Action for Einstein-Cartan?

I don't think it's a trivial problem. One approach might be to take the curvature scalar corresponding to the Levi-Civita connection. When written out in terms of the native connection, this becomes a rather complicated expression involving not just the curvature tensor, but the contorsion and its first derivatives.

Another alternative is to just take the curvature scalar for the native connection and tetrad, itself, as the Lagrangian. The tetrad comes in this quadratically, along with the determinant of the tetrad inverse. The latter gives you a -(metric) x (curvature scalar), the former gives you a +2 (Ricci tensor); when you vary the tetrad.

When you vary the connection, you get the equations relating torsion. For the free field, this results in zero torsion. For coupled fields, I don't know exactly what will happen, since I don't know how the spin tensor is derived from the matter Lagrangian (e.g. is it the variation with respect to the connection?)

In this vein, a different approach was advocated in LNP 107. It differed from Einstein-Cartan in taking the torsion as 0, but the connection as non-metrical. It most interesting feature is that gravity does not enter fundamentally into the picture at all! Instead, the curvature scalar pops up via a partial Legendre transform of the matter Lagrangian.

It might be possible to modify the treatment in LNP 107 to fit Einstein-Cartan; arriving at a similar result. But this requires altering substantial parts of the original text; and the expressions become quite a bit more complex, not explicitly involving torsion.

As to the first idea: I don't know what will happen if you use the curvature scalar for the Levi-Civita tensor expressed in terms of the native connection and [con]torsion.

I've run into a problem similar to that discussed in this thread in the process of trying to implement the idea of gauging a doubly-affine extension of GL(4) that incorporates the Heisenberg algebra for its two generators. What's interesting about this field theory is that for its potentials, you get two sets of "tetrad" fields, whose contraction gives you a rank-2 tensor (i.e., a native "metric") -- however, generally anti-symmetric. There are 16+4+4+1 generators, the field strength corresponding to the last one gives you, essentially, the anti-symmetric part of this "native metric".

It looks a lot like Einstein's old anti-symmetric metric theory ... but now formulated on more solid ground as a gauge theory for this doubly-affine extension of GL(4).

Reference:
LNP 107
Lecture Notes in Physics, 107 (Kijowski, Tulczyjew, 1979)
https://www.amazon.com/dp/0387095381/?tag=pfamazon01-20

 

[Federation 2005]

A Lagrangian for Einstein-Cartan

I looked at the question in some more detail. It turns out that you can build up a Lagrangian from the frame field (e^a), connection one-form (w^{ab}), torsion two-form (t^a) and curvature two-form (R^{ab}).

If you want the Lagrangian to respect gauge-invariance with respect to the connection w^{ab}, this means that derivatives of e^a can only enter into the Lagrangian as gauge-covariant derivatives -- therefore, the functional dependence of the Lagrangian is explicitly on the torsion and frame. Also, dependence on the connection and its derivatives of the connection can only enter through the curvature two-form. Therefore, the Lagrangian is a function of (R^{ab}, t^a and e^a).

To build up a 4-form, using the Minkowski metric eta_{ab} and epsilon_{abcd}, one has only a few combinations. Those of the forms
(1) R^{ab} ^ R_{ab}
(2) epsilon_{abcd} R^{ab} ^ R^{cd}
(3) tau_a ^ tau^a
will either contribute nothing to the field equations or otherwise integrate by parts to a boundary term and combination of one of the forms (4)-(6). The remaining forms are
(4) epsilon_{abcd} e^a ^ e^b ^ R^{cd}
(5) e^a ^ e^b ^ R_{ab}
(6) epsilon_{abcd} e^a ^ e^b ^ e^c ^ e^d.

If a non-zero multiple of either (4) or (5) is present, the field equation for the torsion will be t^a = 0. If (4) is not present, no non-trivial field equation for the curvature will be derivable, therefore a multiple of (4) has to be there.

The term (4) gives you a muliple of the Ricci scalar.

But given a multiple of (4) in the Lagrangian, the torsion will vanish, and the term (5) will reduce to 0 in virtue of the first Bianchi identity (R_{mnrs} + R_{mrsn} + R_{msnr} = 0). Therefore, (5) will not contribute to the field equation for pure gravity. However, it will affect the way matter combines with gravity.

Given that the torsion vanishes, then the field equation derived from (4) will be the usual one you get from the Einstein-Hilbert action.

I think the extra term (5) distinguishes equivalence-principle-violating teleparallel gravity from the teleparallel equivalent of GR. It corresponds to a term of the form epsilon^{mnrs} R_{mnrs}.

The term (6) gives you a contribution proportional to the cosmological constant.

In the presence of matter, an explicit dependence on the connection in the total Lagrangian may occur through the covariant derivatives of the matter fields. Therefore, the total Lagrangian will have a variation of the form

DL = De^a ^ P_a + 1/2 Dw^{ab} ^ S_{ab} - Dt^a ^ T_a - 1/2 DR^{ab} ^ U_{ab}.

You'll end up getting field equations of the form
DT_a = P_a
DU_{ab} = J_{ab} = S_{ab} + T_a ^ e_b - T_b ^ e_a.

The signs on the last two terms may have to be checked. The 2-forms T_a, U_{ab} serve as "superpotentials" for the matter field sources; while the 3-forms P_a, S_{ab} and J_{ab} are the currents for momentum, intrinsic angular momentum and total angular momentum. The relation between S_{ab} and J_{ab} generalizes the spin-orbit decomposition of angular momentum.

An interesting feature of this representation is that when it is applied to pure gravity, there is a corresponding term for P_a. That is, you have a stress tensor for the gravity field! What is it? The Einstein tensor (up to a factor).

In retrospect, this makes sense. If you take the total stress tensor for the Maxwell field, for instance -- adding in the source-potential terms, you get a covariant divergence of 0. Here, not only to you get a covariant divergence of the "total" stress tensor (matter - Einstein tensor) of 0, but the tensor, itself is zero.

Julian Barbour had shown that a careful implementation of the Mach principle via a gauging of the frame field leads to a natural gauge condition ("horizontal stacking") that the total momentum P = 0, and total angular momentum L = 0; as well as a gauging which *defines* the time variable: (Kinetic energy + Gravitational potential = 0).

Interpreting the Einstein tensor as the (minus) stress tensor of the gravity field, then one sees that the field equation, itself, as a generalization of Barbour's conditions to GR.